Erdős (Erdos) Problem #142: status in 2026

Erdős Problem #142

Prove an asymptotic formula for r_k(N), the largest possible size of a subset of 1, …, N that does not contain any non-trivial k-term arithmetic progression. That is, find f_k with r_k(N) / f_k(N) → 1 as N → ∞.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31). Erdős offered a prize of $10000.

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #142 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_142 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #142 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_142`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	4 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-142,
  title        = {Erdős Problem #142},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-142}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Prove an asymptotic formula for $r_{k} (N)$ , the largest possible size of a subset of ${1, \dots, N}$ that does not contain any non-trivial $k$ -term arithmetic progression. That is, find $f_{k}$ with $r_{k} (N) / f_{k} (N) \to 1$ as $N \to \infty$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«142».

theorem erdos_142 (k : ℕ) : (fun N => (r k N : ℝ)) ~[atTop] erdos_142.answer

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/142. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_142.variants.lower — Show that r_k(N) = o_k(N / log N), where r_k(N) the largest possible size of a subset of 1, …, N that does not contain any non-trivial k-term arithmetic…
erdos_142.variants.isTheta — Determine the order of magnitude of r_k(N), the largest possible size of a subset of 1, …, N that does not contain any non-trivial k-term arithmetic…
erdos_142.variants.three — Prove an asymptotic formula for r_3(N), the largest possible size of a subset of 1, …, N that does not contain any non-trivial 3-term arithmetic progression.

References

erdosproblems.com/142

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.